Simulation of topological Zak phase in spin-phononic crystal networks

Topological states of matter are particularly interesting for both fundamental research and practical applications. Simulating topological phases in a quantum system is of great interest due to the ability to explore a plethora of topologically nontrivial phenomena in a controllable fashion. We propose and analyze an efficient scheme for simulating topological Zak phase in two-dimensional spin-phononic crystal networks. We show that through a specially designed periodic driving, one can selectively control and enhance the bipartite silicon-vacancy center arrays, so as to obtain chiral-symmetry-protected spin-spin couplings. More importantly, the Floquet engineering spin-spin interactions support rich quantum phases associated with topological invariants. In momentum space, we analyze and simulate the topological nontrivial properties of the one- and two-dimensional systems. As an application in quantum information processing, we study the robust quantum state transfer via topologically protected edge states. This work opens up new prospects for studying quantum acoustics and offers an experimentally feasible platform for the study of topological phases of matter.

Figure 1

(a) Schematics of the hybrid device studied in this paper. Nanomechanical 2D phonon band-gap setups elaborately designed with high-$Q$ cavities in a patterned diamond membrane. (b) Arrays of SiV color centers are implanted evenly in the phononic waveguide. The lattice constant of the phononic crystal is $a=100$ nm, the sizes of the ellipse holes are $(b,c)=(30,76)$ nm, and the thickness is $t=20$ nm. SiV spins are driven by microwave driving fields. (c) Ground-state energy levels of a single SiV center. Lower and upper states split via an external magnetic field. $g_{n,k}$ describes the coupling strength between the spin and the phonon with the transition $\left(\right|g\rangle \leftrightarrow |f\rangle )$, and ${\mathrm{\Delta }}_{BE}={\omega }_s-{\omega }_{BE}$ is the detuning between the spin transition and phononic band-edge frequency. (d) Displacement pattern of the phononic compression mode at the band-edge frequency ${\omega }_{BE}$.

Figure 2

Schematic diagram of effective spin-spin interactions. (a) Even-neighbor hopping. (b) and (c) illustrate two kinds of odd-neighbor hopping examples, respectively.

Figure 3

(a) Quasienergy spectrum as a function of the driving parameter $a_0$. The corresponding winding number $\mathcal{W}$ is indicated. Here, we assign $|x_j-x_{j+1}|=a$, $L_c=a$. Note that only the first and third neighbor interactions are included. The other parameters are $N=5$, $\omega =10$, $q=1$, and $J_0=1$.

Figure 4

(a1)–(f1) show the dispersion relations for various parameter settings of periodic driving: (a1) $a_0=-16$, (b1) $a_0=-11$, (c1) $a_0=4$, (d1) $a_0=19$, (e1) $a_0=21$, and (f1) $a_0=26$. As the wave number $k$ runs through the Brillouin zone ($-\pi$, $\pi$), the energy spectrum splits into two branches, and there exists a band gap between the lower and higher branches. (a2)–(f2) correspond to the winding configuration of $(d_x,d_y)$ around the origin (red star), and the relevant winding number $\mathcal{W}$ is explicitly shown. In (a2) and (c2), the loop wind avoids the origin, and then $\mathcal{W}=0$. In (b2), (d2), and (f2), the endpoint of $d\left(k\right)$ encircles the origin once, but these are topologically inequivalent. For (d2) and (f2), the endpoint of $d\left(k\right)$ is a closed loop in the counterclockwise direction, and $\mathcal{W}=1$. In contrast, in (b2), the endpoint of $d\left(k\right)$ is along the clockwise direction, and $\mathcal{W}=-1$. In (e2), the endpoint of $d\left(k\right)$ encircles the origin two times, and $\mathcal{W}=2$. Other parameters are the same as those in Fig. 3.

Figure 5

(a1)–(d1) show the energy spectrum in the single-excited state subspace for various parameter settings of periodic driving: (a1) $a_0=-11$, (b1) $a_0=19$, (c1) $a_0=21$, and (d1) $a_0=26$. For the cases with $\mathcal{W}=1$ and $\mathcal{W}=-1$, there are two zero-energy eigenvalues. For the case with $\mathcal{W}=2$, there are four zero-energy eigenvalues. The zero-energy eigenvalues (the red points) correspond to the topological edge states. The rest of the eigenvalues (the blue points) correspond to the bulk states of the system. (a2)–(d2) show the related eigenfunction of the gapless modes. Here, we consider $N=50$. Other parameters are the same as those in Fig. 3.

Figure 6

Schematic diagram of the 2D Floquet engineering spin-spin interaction. There are four spins in a unit cell, which are labeled as {$A,B,C,D$}, respectively. Here, we introduce $(n,m)$ to describe the position of each unit cell in the 2D spin-spin networks. For simplicity, only the nearest-neighbor interactions are illustrated.

Figure 7

(a) Band structure of the 2D Floquet engineering spin-spin interaction in the $\mathbf{k}$ space, $a_0=4$. The energy spectrum has four branches. The lowest and highest bands are isolated, while the two middle bands are connected at the edges of the Brillouin zone: (0,0), ($\pm \pi ,\pm \pi$), and ($\mp \pi ,\pm \pi$). (b) Dispersion relation along the path $M\to \mathrm{\Gamma }\to X\to M$ in the first Brillouin zone, $a_0=21$. The parities of each band at high-symmetry points are marked by the plus and minus signs. (c) The bulk spectra for $a_0=5$, which describes a topological phase transition. Other parameters are the same as those in Fig. 3.

Figure 8

Projected band structures for various parameter settings of periodic driving: (a) $a_0=4$, (b) $a_0=-11$, and (c) $a_0=21$. The blue and red curves denote the bulk and edge modes, respectively. Here, we consider $N_x=11$. Other parameters are the same as those in Fig. 3.

Figure 9

Excitation dynamics of the left end spin for various parameter settings of periodic driving: (a) $a_0=12$, (b) $a_0=24$, and (c) $a_0=-2$. In (a), we also add the result in the case of ${\gamma }_s=1\times {10}^{-4}{J}_0$. Here, we consider $N=3$. Other parameters are the same as those in Fig. 3.